Mathematics Commons^™

Cycle Equivalence Of Graph Dynamical Systems, Matthew Macauley, Henning S. Mortveit

Publications

Graph dynamical systems (GDSs) generalize concepts such as cellular automata and Boolean networks and can describe a wide range of distributed, nonlinear phenomena. Two GDSs are cycle equivalent if their periodic orbits are isomorphic as directed graphs, which captures the notion of having comparable long-term dynamics. In this paper, we study cycle equivalence of GDSs in which the vertex functions are applied sequentially through an update sequence. The main result is a general characterization of cycle equivalence based on the underlying graph Y and the update sequences. We construct and analyse two graphs C(Y) and D( …

Third Grade Students' Challenges And Strategies To Solving Mathematical Word Problems, Elizabeth Bernadette

Open Access Theses & Dissertations

This project explores the difficulties and challenges that third grade students face solving mathematical word problems. Three students were asked to be participants and share their knowledge on this topic as well as their work. After extensive interviews it was concluded that the challenges of mathematical word problems include but are not limited to the level of reading comprehension, conceptual understanding of mathematical concepts, and the belief that math is a compilation of computations and unexplainable procedures. The participants provided insight as to what strategies are helpful to students. These strategies include group discussion on problem solving strategies, self-assessment, incorporating …

Linear Operators That Preserve The Edgesum Of A Graph, Ian June L. Garces, Siegfred Alan C. Baluyot

Mathematics Faculty Publications

No abstract provided.

Gutt Products And Representations Of Lie Groups, Job A. Nable

Mathematics Faculty Publications

No abstract provided.

Generalized Q-Functions And Dirichlet-To-Neumann Maps For Elliptic Differential Operators, Daniel Alpay, Jussi Behrndt

Mathematics, Physics, and Computer Science Faculty Articles and Research

The classical concept of Q-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be interpreted as a generalized Q-function. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein type formulas for the difference of the resolvents and trace formulas in an H2-framework are obtained.

Counting Links In Complete Graphs, Thomas Fleming, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

We find the minimal number of non-trivial links in an embedding of any complete kk-partite graph on 7 vertices (including K₇, which has at least 21 non-trivial links). We give either exact values or upper and lower bounds for the minimal number of non-trivial links for all complete kk-partite graphs on 8 vertices. We also look at larger complete bipartite graphs, and state a conjecture relating minimal linking embeddings with minimal book embeddings.

Ecosystem Modeling Of College Drinking: Parameter Estimation And Comparing Models To Data, Azmy S. Ackleh, Ben G. Fitzpatrick, Richard Scribner, Neal Simonsen, Jeremy J. Thibodeaux

Mathematics, Statistics and Data Science Faculty Works

Recently we developed a model composed of five impulsive differential equations that describes the changes in drinking patterns (that persist at epidemic level) amongst college students. Many of the model parameters cannot be measured directly from data; thus, an inverse problem approach, which chooses the set of parameters that results in the “best” model to data fit, is crucial for using this model as a predictive tool. The purpose of this paper is to present the procedure and results of an unconventional approach to parameter estimation that we developed after more common approaches were unsuccessful for our specific problem. The …

Eigenvalue Inequalities For A Family Of Spherically Symmetric Riemannian Manifolds, Julie Miker

University of Kentucky Doctoral Dissertations

This thesis considers two isoperimetric inequalities for the eigenvalues of the Laplacian on a family of spherically symmetric Riemannian manifolds. The Payne-Pólya-Weinberger Conjecture (PPW) states that for a bounded domain Ω in Euclidean space Rⁿ, the ratio λ1(Ω)/λ0(Ω) of the first two eigenvalues of the Dirichlet Laplacian is bounded by the corresponding eigenvalue ratio for the Dirichlet Laplacian on the ball B_Ωof equal volume. The Szegö-Weinberger inequality states that for a bounded domain Ω in Euclidean space Rⁿ, the first nonzero eigenvalue of the Neumann Laplacian μ1(Ω) is maximized on the ball B_{Ω …}

Rook Polynomials In Higher Dimensions, Nicholas Krzywonos, Feryal Alayont

Student Summer Scholars Manuscripts

A rook polynomial counts the number of placements of non-attacking rooks on a board. In this paper we describe generalizations of the definition and properties of rook polynomials to "boards" in three and higher dimensions. We also defefine generalizations of special two dimensional boards to three dimensions, including the triangle board and the board representing the probleme des rencontres. The number of rook placements on these three dimensional families of rook boards are shown to be related to famous number sequences, such as central factorial numbers, the number of Latin rectangles and the Genocchi numbers.

Examples Of Hyperbolic Knots With Distance 3 Toroidal Surgeries In The 3-Sphere, Cesar Garza

Open Access Theses & Dissertations

By the work of Thurston, any surgery on a hyperbolic knot in the 3-sphere produces a hyperbolic 3-manifold except in at most finitely many cases. So far, the figure-8 knot seems to be the best candidate for a hyperbolic knot with the most (8) non-trivial exceptional surgeries. In recent years, much progress has been made in the classification of hyperbolic knots admitting more than one exceptional toroidal surgery. In fact, such classification is known for toroidal surgeries with distance at least 4.

We give a classification of hyperbolic knots in $S^3$ admitting two toroidal surgeries at distance 3, whose slopes …

Semi-Automated Frame Transformations Using Fft Analysis On 2-D Images, Francisco Javier Osuna

Open Access Theses & Dissertations

Cassini entered Saturn's orbit on July 1, 2004 beginning a four-year exploration of Saturn. In 2008 the mission was extended, and Cassini continues to collect and transmit images and data collected during its mission. In order to accurately interpret images, it is necessary to know the location and orientation of the camera provided the field of view when the image was collected. While the mission managers provide initial estimates of this orientation, scientific analysis requires better estimates than the initial data provided. Navigation is a process for improving the estimation of the true camera pointing vector as determined by features …

Shadowing Chaos Via Optimization, Henrik Haakonsen

Mathematics, Statistics, and Computer Science Honors Projects

A prominent idea in the theory of chaos is that of shadowing, which says that, in many cases, the numerical results one sees after accuracy is lost are not total nonsense, but are in fact very close to the exact trajectory for an initial value that is near the one used. Using high-precision computation, I have researched the use of optimization as a way of finding exact shadows for several chaotic systems, such as the quadratic map r x (1 - x) and a billiard problem from the SIAM 100-Digit Challenge.

On The Estimation Of Averages Over Infinite Intervals With An Application To Average Persistence In Population Models, Sean F. Ellermeyer

Faculty Articles

We establish a general result for estimating the upper average of a continuous and bounded function over an infinite interval. As an application, we show that a previously studied model of microbial growth in a chemostat with time–varying nutrient input admits solutions (populations) that exhibit weak persistence but not weak average persistence.

Proceedings Of The Third Annual Meeting Of The Georgia Association Of Mathematics Teacher Educators Front Matter

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Contents of 3rd Annual GAMTE Proceedings Front Matter:

• Proceedings Committee
• Officers of GAMTE
• Purposes and Goals of GAMTE
• Table of Contents
• Letter from President

Analysis Of Achievement For Understanding Geometry, Annita W. Hunt

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

The purpose of this study was to investigate the effectiveness of a mathematics professional development course. More specifically, in this study we examine whether geometric experiences have an impact on level of performance in mathematics. The van Hiele (Fuys, D., Geddes, D., & Tischler, R., 1988) model of geometric understanding provided a research framework from which to view geometric understanding. This model suggests five levels of understanding that should be taken into consideration when examining levels of geometric thinking: Visual, Descriptive/Analytic, Abstract/Relational, Formal Deduction/Proof, and Rigor. The sample under study was three cohorts of practicing elementary teachers and mathematics coaches …

Iterative Methods For Computing Eigenvalues And Exponentials Of Large Matrices, Ping Zhang

University of Kentucky Doctoral Dissertations

In this dissertation, we study iterative methods for computing eigenvalues and exponentials of large matrices. These types of computational problems arise in a large number of applications, including mathematical models in economics, physical and biological processes. Although numerical methods for computing eigenvalues and matrix exponentials have been well studied in the literature, there is a lack of analysis in inexact iterative methods for eigenvalue computation and certain variants of the Krylov subspace methods for approximating the matrix exponentials. In this work, we proposed an inexact inverse subspace iteration method that generalizes the inexact inverse iteration for computing multiple and clustered …

Gorenstein Flat Dimension Of Complexes, Alina Iacob

Mathematical Sciences: Faculty Publications

We define a notion of Gorenstein flat dimension for unbounded complexes over left GF-closed rings. Over Gorenstein rings we introduce a notion of Gorenstein cohomology for complexes; we also define a generalized Tate cohomology for complexes over Gorenstein rings, and we show that there is a close connection between the absolute, the Gorenstein and the generalized Tate cohomology.

On Some Differential Equations, Mekki Terbeche, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

This paper investigates Cauchy and Goursat problems for partial differential operators. Successive approximation techniques for partial differential equations and the estimated results are employed to obtain the existence and the uniqueness of the solutions of such problems. An extended Darboux-Goursat-Beudon problem is studied.

[Introduction To] The Hardy Space Of A Slit Domain, William T. Ross, Alexandra Aleman, Nathan S. Feldman

Bookshelf

If H is a Hilbert space and T : H ? H is a continous linear operator, a natural question to ask is: What are the closed subspaces M of H for which T M ? M? Of course the famous invariant subspace problem asks whether or not T has any non-trivial invariant subspaces. This monograph is part of a long line of study of the invariant subspaces of the operator T = M (multiplication by the independent variable z, i. e. , M f = zf ) on a z z Hilbert space of analytic functions on a bounded …

Dynamics And Rheology Of Biaxial Liquid Crystal Polymers, Sarthok K. Sircar

Theses and Dissertations

In this thesis we derive a hydrodynamical kinetic theory to study the orientational response of a mesoscopic system of nematic liquid crystals in the presence of an external flow field. Various problems have been attempted in this direction. First, we understand the steady-state behavior of uniaxial LCPs under an imposed elongational flow, electric and magnetic field respectively. We show that (1) the Smoluchowski equation can be cast into a generic form, (2) the external field is parallel to one of the eigenvectors of the second moment tensor, and (3) the steady state probability density function is of the Boltzmann type. …

Gas-Kinetic Schemes For Direct Numerical Simulations Of Compressible Homogeneous Turbulence, Wei Liao, Yan Peng, Li-Shi Luo

Mathematics & Statistics Faculty Publications

We apply the gas-kinetic scheme (GKS) for the direct numerical simulations (DNSs) of compressible decaying homogeneous isotropic turbulence (DHIT). We intend to study the accuracy, stability, and efficiency of the gas-kinetic scheme for DNS of compressible homogeneous turbulence depending on both flow conditions and numerics. In particular, we study the GKS with multidimensional, quasi-one-dimensional, dimensional-splitting, and smooth-flow approximations. We simulate the compressible DHIT with the Taylor microscale Reynolds number Reλ =72.0 and the turbulence Mach number Ma_t between 0.1 and 0.6. We compute the low-order statistical quantities including the total kinetic energy K (t), the dissipation rate ε (t), …

Direct Products And The Intersection Map Of Certain Classes Of Finite Groups, Julia Chifman

University of Kentucky Doctoral Dissertations

The main goal of this work is to examine classes of finite groups in which normality, permutability and Sylow-permutability are transitive relations. These classes of groups are called T , PT and PST , respectively. The main focus is on direct products of T , PT and PST groups and the behavior of a collection of cyclic normal, permutable and Sylow-permutable subgroups under the intersection map. In general, a direct product of finitely many groups from one of these classes does not belong to the same class, unless the orders of the direct factors are relatively prime. Examples suggest that …

The Generalized Burnside And Representation Rings, Eric B. Kahn

University of Kentucky Doctoral Dissertations

Making use of linear and homological algebra techniques we study the linearization map between the generalized Burnside and rational representation rings of a group G. For groups G and H, the generalized Burnside ring is the Grothendieck construction of the semiring of G × H-sets with a free H-action. The generalized representation ring is the Grothendieck construction of the semiring of rational G×H-modules that are free as rational H-modules. The canonical map between these two rings mapping the isomorphism class of a G-set X to the class of its permutation module …

Testing The Fixed Effects Restrictions? A Monte Carlo Study Of Chamberlain's Minimum Chi-Squared Test, Badi H. Baltagi, Georges Bresson, Alain Pirotte

Center for Policy Research

Chamberlain (1982) showed that the fixed effects (FE) specification imposes testable restrictions on the coefficients from regressions of all leads and lags of dependent variables on all leads and lags of independent variables. Angrist and Newey (1991) suggested computing this test statistic as the degrees of freedom times the R2 from a regression of within residuals on all leads and lags of the exogenous variables. Despite the simplicity of these tests, they are not commonly used in practice. Instead, a Hausman (1978) test is used based on a contrast of the fixed and random effects specifications. We advocate the use …

Convergence And The Lebesgue Integral, Ryan Vail Thomas

Graduate Theses/Dissertations

In this paper, we examine the theory of integration of functions of real variables. Background information in measure theory and convergence is provided and several examples are considered. We compare Riemann and Lebesgue integration and develop several important theorems. In particular, the Monotone Convergence Theorem and Dominated Convergence Theorem are considered under both pointwise convergence and convergence in measure.

Ambigrams: The Art And The Math, Callie Kronlage

Honors Program Theses

Because ambigrams are a relatively new concept, I found little mathematical research on them. Throughout this paper, I will address the role of ambigrams as a mathematical entity, an artistic outlet, and even an educational tool. Along with ways to use ambigrams, there are also descriptions of the nine different types of ambigrams, and a chart that can be used to create two of the most common types of ambigrams. Mathematically, ambigrams contain many symmetrical aspects, which are important to the study of Geometry. Within this, I have researched mathematical aspects of three of the ambigram types and found some …

The Number Of Permutations Realized By A Shift, Sergi Elizalde

Dartmouth Scholarship

A permutation $\pi$ is realized by the shift on N symbols if there is an infinite word on an N-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as $\pi$. The set of realized permutations is closed under consecutive pattern containment. Permutations that cannot be realized are called forbidden patterns. It was shown in [J. M. Amigó, S. Elizalde, and M. B. Kennel, J. Combin. Theory Ser. A, 115 (2008), pp. 485–504] that the shortest forbidden patterns of the shift on N symbols have length $N+2$. In this paper we give …

Standing Waves Of Spatially Discrete Fitzhugh-Nagumo Equations, Joseph Segal

Electronic Theses and Dissertations

We study a system of spatially discrete FitzHugh-Nagumo equations, which are nonlinear differential-difference equations on an infinite one-dimensional lattice. These equations are used as a model of impulse propagation in nerve cells. We employ McKean's caricature of the cubic as our nonlinearity, which allows us to reduce the nonlinear problem into a linear inhomogeneous problem. We find exact solutions for standing waves, which are steady states of the system. We derive formulas for all 1-pulse solutions. We determine the range of parameter values that allow for the existence of standing waves. We use numerical methods to demonstrate the stability of …

The Sheffer B-Type 1 Orthogonal Polynomial Sequences, Daniel Galiffa

Electronic Theses and Dissertations

In 1939, I.M. Sheffer proved that every polynomial sequence belongs to one and only one type. Sheffer extensively developed properties of the B-Type 0 polynomial sequences and determined which sets are also orthogonal. He subsequently generalized his classification method to the case of arbitrary B-Type k by constructing the generalized generating function A(t)exp[xH1(t) + · · · + xk+1Hk(t)] = ∑∞n=0 Pn(x)tn, with Hi(t) = hi,iti + hi,i+1t i+1 + · · · , h1,1 ≠ 0. Although extensive research has been done on characterizing polynomial sequences, no analysis has yet been completed on sets of type one or higher …

Optimal Dual Frames For Erasures And Discrete Gabor Frames, Jerry Lopez

Electronic Theses and Dissertations

Since their discovery in the early 1950's, frames have emerged as an important tool in areas such as signal processing, image processing, data compression and sampling theory, just to name a few. Our purpose of this dissertation is to investigate dual frames and the ability to find dual frames which are optimal when coping with the problem of erasures in data transmission. In addition, we study a special class of frames which exhibit algebraic structure, discrete Gabor frames. Much work has been done in the study of discrete Gabor frames in Rn, but very little is known about the l2(Z) …